Generally, a volume flow rate can be defined as a volume of a fluid flowing through a flow passage cross-section within a time unit. For example, a flow rate Q of a fluid flowing through a flow passage pipe arrangement 1 illustrated in FIG. 1 is represented as an integrated value of normal-directional components with respect to a flow passage cross-section of flow velocities at respective points on the flow passage cross-section, as follows:
                                                                                          Q                  =                                    ⁢                                                            ∫                      S                                        ⁢                                                                                  ⁢                                          ds                      ⁢                                                                                          ⁢                                              v                        z                                                                                                                                                                  =                                    ⁢                                                            ∫                                              x                        min                                                                    x                        max                                                              ⁢                                          dx                      ⁢                                                                        ∫                                                                                    y                              min                                                        ⁡                                                          (                              x                              )                                                                                                                                          y                              max                                                        ⁡                                                          (                              x                              )                                                                                                      ⁢                                                  dy                          ⁢                                                                                                          ⁢                                                                                    v                              z                                                        ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                                                                                                                                                                                =                                    ⁢                                                            ∫                                              x                        min                                                                    x                        max                                                              ⁢                                                                                  ⁢                                          dx                      ⁢                                                                                          ⁢                                                                        V                          z                                                ⁡                                                  (                          x                          )                                                                    ⁢                                              L                        ⁡                                                  (                          x                          )                                                                                                                                                  ⁢                                          ⁢                      where            ,                          ⁢                                                      (        1        )                                                      V                          z              ⁢                                                                            ⁡                      (            x            )                          =                                            ∫                                                y                  min                                ⁡                                  (                  x                  )                                                                              y                  max                                ⁡                                  (                  x                  )                                                      ⁢                          dy              ⁢                                                          ⁢                                                v                                      z                    ⁢                                                                                                                ⁡                                  (                                      x                    ,                    y                                    )                                                                                                        y                max                            ⁡                              (                x                )                                      -                                          y                min                            ⁡                              (                x                )                                                                        (        2        )                                          L          ⁢                      (            x            )                          =                                            y              max                        ⁡                          (              x              )                                -                                                    y                min                            ⁡                              (                x                )                                      .                                              (        3        )            
In the Formulas (1) to (3), S represents a flow passage cross-section, wherein a position in the flow passage cross-section S is represented by (x, y). vz represents a normal-directional (z-directional) component of a flow velocity of the fluid with respect to the flow passage cross-section S, and vz(x, y) represents a normal-directional component at a position (x, y) of a flow velocity of the fluid at the position (x, y) on the flow passage cross-section S. ymin(x) and ymax(x) represent, respectively, a minimum value and a maximum value of the y-coordinate when the x-coordinate is fixed on the flow passage cross-section S, and xmin and xmax represent, respectively, a minimum value and a maximum value of the x-coordinate on the flow passage cross-section S. Additionally, in FIG. 1, the flow passage cross-section S1 is a plane orthogonal to an axis of the flow passage pipe arrangement 1, but even if the flow passage cross-section is inclined with respect to the axis, or is a curved surface, the flow rate can be represented by the above Formulas (1) to (3).
As is clear from the above Formula (2), when xi as the x-coordinate is set to a certain point, Vz(xi) represents an average of values of vz(x, y) for respective points (x, y), on a region in the flow passage cross-section S in which the x-coordinate is equal to xi, namely on a line segment AB (FIG. 2), and the Vz(xi) can be measured by, for example, an ultrasonic wave method.
In one example, as illustrated in FIG. 3, assume that two points each shifted from a respective one of points A, B in a z-direction by a given distance are represented, respectively, as A* and B*. In this case, by utilizing a phenomenon that an apparent difference in sound velocity corresponding to flow velocity occurs between when an ultrasonic wave is propagated in a direction from the point A* to the point B* and when the ultrasonic wave is propagated in a direction from the point B* to the point A*, an average of values of vz(x, y) between A* and B* can be derived from respective propagation times in the two directions. Generally, the average value can be deemed as an average flow velocity Vz(xi) on the line segment AB.
More specifically, as illustrated in FIG. 4, an ultrasonic element 2a configured to transmit an ultrasonic wave from the point A* and an ultrasonic element 2b configured to transmit an ultrasonic wave from the point B* are disposed on the flow passage pipe arrangement 1, to measure a propagation time of the ultrasonic wave transmitted from the ultrasonic element 2a until it is received by the ultrasonic element 2b, and a propagation time of the ultrasonic wave transmitted from the ultrasonic element 2b until it is received by the ultrasonic element 2a. In a simple example, as described in the Patent Document 1, on an assumption that a propagation time of the ultrasonic wave transmitted from the ultrasonic element 2a to the ultrasonic element 2b until it reaches the ultrasonic element 2b via a path having a length L, and a propagation time of the ultrasonic wave transmitted from the ultrasonic element 2b to the ultrasonic element 2a until it reaches the ultrasonic element 2a via the path are represented, respectively, as t1 and t2, the average of the values of vz(x, y) can be determined by the following Formula (4) (where an angle between the ultrasonic wave transmission and receipt path and the z-direction is represented as θ (theta)).
                                                        v              z                        ⁡                          (                              x                ,                y                            )                                _                =                              L                          2              ⁢                                                          ⁢              cos              ⁢                                                          ⁢              θ                                ⁢                      (                                          1                                  t                  2                                            -                              1                                  t                  1                                                      )                                              (        4        )            
Generally, this average value can be used as a value of Vz(xi).
In practice, Vz(xi) can be measured at discrete positions. Thus, in many cases, it is necessary to calculate an approximate value of Q using numerical integration. Further, in the case of performing measurement using the pair of ultrasonic elements as illustrated in FIG. 4, it is conceivable to arrange the pairs of ultrasonic elements on the flow passage pipe arrangement in a number corresponding to the number of xi points to be measured. In this case, however, there is a limit on the number of measurement points which can be arranged, and thus it is necessary to use a calculation technique capable of maintaining good accuracy even under a small number of measurement points.
A method for deriving an approximate value of the flow rate Q represented by the Formula (1), from the discretely measured flow velocities, by Gaussian quadrature (Gauss-Legendre type, Chebyshev type, and Lobatto type) is disclosed in the Patent Document 2. The method disclosed in the Patent Document 2 is configured to determine an approximate value QN of the flow rate Q by the following Formula, using Gaussian quadrature under a condition that an integrand is set to Vz(x)L(x) and a weighting function is set to 1 (constant):
                              Q          N                =                              ∑                          i              =              1                        N                    ⁢                                          ⁢                                    w              i                        ⁢                                          V                z                            ⁡                              (                                  x                  i                                )                                      ⁢                                          L                ⁡                                  (                                      x                    i                                    )                                            .                                                          (        5        )            In this Formula, xi and wi represent, respectively, a node and a coefficient of the Gaussian quadrature. The Formula (5) is calculated using discrete flow velocity values of Vz(xi) measured at respective nodes of x=xi.
The Patent Documents 3 and 4 disclose a method for deriving an approximate value of flow rate of a fluid flowing through a flow passage of a circular pipe having a radius R, from a discretely measured flow velocities. In the flow passage of the circular pipe having a radius R, when the x-coordinate is defined on a flow passage cross-section, L(x) in the Formula (3) is expressed as follows:
                              L          ⁡                      (            x            )                          =                  2          ⁢          R          ⁢                                                    1                -                                                      (                                          x                      R                                        )                                    2                                                      .                                              (        6        )            Utilizing this, in the methods disclosed in the Patent Documents 3 and 4, on an assumption that the weighting function is set as follows:
                                          1            -                                          (                                  x                  R                                )                            2                                      ,                            (        7        )            and on an assumption that the integrand is set to 2RVz(x), the nodes xi and the coefficients wi are calculated in accordance with Gaussian quadrature (Gauss-Chebyshev type). The approximate value QN of the flow rate Q is determined by the following approximation formula, using discrete values of the flow velocity Vz(xi) measured at respective nodes of x=xi;
                              Q          N                =                              ∑                          i              =              1                        N                    ⁢                                    w              i                        ⁢            2            ⁢                                                            RV                  z                                ⁡                                  (                                      x                    i                                    )                                            .                                                          (        8        )            
In the above conventional techniques, the flow velocity distribution itself is included in the integrand. Thus, there is a problem that, for example, in a situation where the flow velocity distribution is largely distorted, a higher-order quadrature (namely, a larger number of measurement points) is required to obtain an accurate approximate value.